Bounded linear maps #
This file defines a class stating that a map between normed vector spaces is (bi)linear and
continuous.
Instead of asking for continuity, the definition takes the equivalent condition (because the space
is normed) that ‖f x‖ is bounded by a multiple of ‖x‖. Hence the "bounded" in the name refers to
‖f x‖/‖x‖ rather than ‖f x‖ itself.
Main definitions #
IsBoundedLinearMap: Class stating that a mapf : E → Fis linear and has‖f x‖bounded by a multiple of‖x‖.IsBoundedBilinearMap: Class stating that a mapf : E × F → Gis bilinear and continuous, but through the simpler to provide statement that‖f (x, y)‖is bounded by a multiple of‖x‖ * ‖y‖IsBoundedBilinearMap.linearDeriv: Derivative of a continuous bilinear map as a linear map.IsBoundedBilinearMap.deriv: Derivative of a continuous bilinear map as a continuous linear map. The proof that it is indeed the derivative isIsBoundedBilinearMap.hasFDerivAtinAnalysis.Calculus.FDeriv.
Main theorems #
IsBoundedBilinearMap.continuous: A bounded bilinear map is continuous.ContinuousLinearEquiv.isOpen: The continuous linear equivalences are an open subset of the set of continuous linear maps between a pair of Banach spaces. Placed in this file because its proof usesIsBoundedBilinearMap.continuous.
Notes #
The main use of this file is IsBoundedBilinearMap. The file Analysis.NormedSpace.Multilinear
already expounds the theory of multilinear maps, but the 2-variables case is sufficiently simpler
to currently deserve its own treatment.
IsBoundedLinearMap is effectively an unbundled version of ContinuousLinearMap (defined
in Topology.Algebra.Module.Basic, theory over normed spaces developed in
Analysis.NormedSpace.OperatorNorm), albeit the name disparity. A bundled
ContinuousLinearMap is to be preferred over an IsBoundedLinearMap hypothesis. Historical
artifact, really.
structure
IsBoundedLinearMap
(𝕜 : Type u_5)
[NormedField 𝕜]
{E : Type u_6}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_7}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
(f : E → F)
extends
IsLinearMap
:
A function f satisfies IsBoundedLinearMap 𝕜 f if it is linear and satisfies the
inequality ‖f x‖ ≤ M * ‖x‖ for some positive constant M.
Instances For
theorem
IsLinearMap.with_bound
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(hf : IsLinearMap 𝕜 f)
(M : ℝ)
(h : ∀ (x : E), ‖f x‖ ≤ M * ‖x‖)
:
theorem
ContinuousLinearMap.isBoundedLinearMap
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
(f : E →L[𝕜] F)
:
IsBoundedLinearMap 𝕜 ↑f
A continuous linear map satisfies IsBoundedLinearMap
def
IsBoundedLinearMap.toLinearMap
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
(f : E → F)
(h : IsBoundedLinearMap 𝕜 f)
:
Construct a linear map from a function f satisfying IsBoundedLinearMap 𝕜 f.
Equations
- IsBoundedLinearMap.toLinearMap f h = IsLinearMap.mk' f (_ : IsLinearMap 𝕜 f)
Instances For
def
IsBoundedLinearMap.toContinuousLinearMap
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
:
Construct a continuous linear map from IsBoundedLinearMap.
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
IsBoundedLinearMap.zero
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
:
IsBoundedLinearMap 𝕜 fun x => 0
theorem
IsBoundedLinearMap.id
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
:
IsBoundedLinearMap 𝕜 fun x => x
theorem
IsBoundedLinearMap.fst
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
:
IsBoundedLinearMap 𝕜 fun x => x.fst
theorem
IsBoundedLinearMap.snd
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
:
IsBoundedLinearMap 𝕜 fun x => x.snd
theorem
IsBoundedLinearMap.smul
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(c : 𝕜)
(hf : IsBoundedLinearMap 𝕜 f)
:
IsBoundedLinearMap 𝕜 (c • f)
theorem
IsBoundedLinearMap.neg
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
:
IsBoundedLinearMap 𝕜 fun e => -f e
theorem
IsBoundedLinearMap.add
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
{g : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
(hg : IsBoundedLinearMap 𝕜 g)
:
IsBoundedLinearMap 𝕜 fun e => f e + g e
theorem
IsBoundedLinearMap.sub
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
{g : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
(hg : IsBoundedLinearMap 𝕜 g)
:
IsBoundedLinearMap 𝕜 fun e => f e - g e
theorem
IsBoundedLinearMap.comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E → F}
{g : F → G}
(hg : IsBoundedLinearMap 𝕜 g)
(hf : IsBoundedLinearMap 𝕜 f)
:
IsBoundedLinearMap 𝕜 (g ∘ f)
theorem
IsBoundedLinearMap.tendsto
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(x : E)
(hf : IsBoundedLinearMap 𝕜 f)
:
Filter.Tendsto f (nhds x) (nhds (f x))
theorem
IsBoundedLinearMap.continuous
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
:
theorem
IsBoundedLinearMap.lim_zero_bounded_linear_map
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(hf : IsBoundedLinearMap 𝕜 f)
:
Filter.Tendsto f (nhds 0) (nhds 0)
theorem
IsBoundedLinearMap.isBigO_id
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(h : IsBoundedLinearMap 𝕜 f)
(l : Filter E)
:
theorem
IsBoundedLinearMap.isBigO_comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{E : Type u_5}
{g : F → G}
(hg : IsBoundedLinearMap 𝕜 g)
{f : E → F}
(l : Filter E)
:
theorem
IsBoundedLinearMap.isBigO_sub
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{f : E → F}
(h : IsBoundedLinearMap 𝕜 f)
(l : Filter E)
(x : E)
:
theorem
isBoundedLinearMap_prod_multilinear
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{ι : Type u_5}
[Fintype ι]
{E : ι → Type u_6}
[(i : ι) → NormedAddCommGroup (E i)]
[(i : ι) → NormedSpace 𝕜 (E i)]
:
IsBoundedLinearMap 𝕜 fun p => ContinuousMultilinearMap.prod p.fst p.snd
Taking the cartesian product of two continuous multilinear maps is a bounded linear operation.
theorem
isBoundedLinearMap_continuousMultilinearMap_comp_linear
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{ι : Type u_5}
[Fintype ι]
(g : G →L[𝕜] E)
:
IsBoundedLinearMap 𝕜 fun f => ContinuousMultilinearMap.compContinuousLinearMap f fun x => g
Given a fixed continuous linear map g, associating to a continuous multilinear map f the
continuous multilinear map f (g m₁, ..., g mₙ) is a bounded linear operation.
We prove some computation rules for continuous (semi-)bilinear maps in their first argument.
If f is a continuous bilinear map, to use the corresponding rules for the second argument, use
(f _).map_add and similar.
We have to assume that F and G are normed spaces in this section, to use
ContinuousLinearMap.toNormedAddCommGroup, but we don't need to assume this for the first
argument of f.
theorem
ContinuousLinearMap.map_add₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{R : Type u_5}
{𝕜₂ : Type u_6}
{𝕜' : Type u_7}
[NontriviallyNormedField 𝕜']
[NontriviallyNormedField 𝕜₂]
{M : Type u_8}
[TopologicalSpace M]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{G' : Type u_9}
[NormedAddCommGroup G']
[NormedSpace 𝕜₂ G']
[NormedSpace 𝕜' G']
[SMulCommClass 𝕜₂ 𝕜' G']
[Semiring R]
[AddCommMonoid M]
[Module R M]
{ρ₁₂ : R →+* 𝕜'}
(f : M →SL[ρ₁₂] F →SL[σ₁₂] G')
(x : M)
(x' : M)
(y : F)
:
theorem
ContinuousLinearMap.map_zero₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{R : Type u_5}
{𝕜₂ : Type u_6}
{𝕜' : Type u_7}
[NontriviallyNormedField 𝕜']
[NontriviallyNormedField 𝕜₂]
{M : Type u_8}
[TopologicalSpace M]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{G' : Type u_9}
[NormedAddCommGroup G']
[NormedSpace 𝕜₂ G']
[NormedSpace 𝕜' G']
[SMulCommClass 𝕜₂ 𝕜' G']
[Semiring R]
[AddCommMonoid M]
[Module R M]
{ρ₁₂ : R →+* 𝕜'}
(f : M →SL[ρ₁₂] F →SL[σ₁₂] G')
(y : F)
:
↑(↑f 0) y = 0
theorem
ContinuousLinearMap.map_smulₛₗ₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{R : Type u_5}
{𝕜₂ : Type u_6}
{𝕜' : Type u_7}
[NontriviallyNormedField 𝕜']
[NontriviallyNormedField 𝕜₂]
{M : Type u_8}
[TopologicalSpace M]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{G' : Type u_9}
[NormedAddCommGroup G']
[NormedSpace 𝕜₂ G']
[NormedSpace 𝕜' G']
[SMulCommClass 𝕜₂ 𝕜' G']
[Semiring R]
[AddCommMonoid M]
[Module R M]
{ρ₁₂ : R →+* 𝕜'}
(f : M →SL[ρ₁₂] F →SL[σ₁₂] G')
(c : R)
(x : M)
(y : F)
:
theorem
ContinuousLinearMap.map_sub₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{R : Type u_5}
{𝕜₂ : Type u_6}
{𝕜' : Type u_7}
[NontriviallyNormedField 𝕜']
[NontriviallyNormedField 𝕜₂]
{M : Type u_8}
[TopologicalSpace M]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{G' : Type u_9}
[NormedAddCommGroup G']
[NormedSpace 𝕜₂ G']
[NormedSpace 𝕜' G']
[SMulCommClass 𝕜₂ 𝕜' G']
[Ring R]
[AddCommGroup M]
[Module R M]
{ρ₁₂ : R →+* 𝕜'}
(f : M →SL[ρ₁₂] F →SL[σ₁₂] G')
(x : M)
(x' : M)
(y : F)
:
theorem
ContinuousLinearMap.map_neg₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{R : Type u_5}
{𝕜₂ : Type u_6}
{𝕜' : Type u_7}
[NontriviallyNormedField 𝕜']
[NontriviallyNormedField 𝕜₂]
{M : Type u_8}
[TopologicalSpace M]
{σ₁₂ : 𝕜 →+* 𝕜₂}
{G' : Type u_9}
[NormedAddCommGroup G']
[NormedSpace 𝕜₂ G']
[NormedSpace 𝕜' G']
[SMulCommClass 𝕜₂ 𝕜' G']
[Ring R]
[AddCommGroup M]
[Module R M]
{ρ₁₂ : R →+* 𝕜'}
(f : M →SL[ρ₁₂] F →SL[σ₁₂] G')
(x : M)
(y : F)
:
theorem
ContinuousLinearMap.map_smul₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(f : E →L[𝕜] F →L[𝕜] G)
(c : 𝕜)
(x : E)
(y : F)
:
structure
IsBoundedBilinearMap
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(f : E × F → G)
:
A map f : E × F → G satisfies IsBoundedBilinearMap 𝕜 f if it is bilinear and
continuous.
Instances For
theorem
ContinuousLinearMap.isBoundedBilinearMap
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(f : E →L[𝕜] F →L[𝕜] G)
:
IsBoundedBilinearMap 𝕜 fun x => ↑(↑f x.fst) x.snd
def
IsBoundedBilinearMap.toContinuousLinearMap
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(hf : IsBoundedBilinearMap 𝕜 f)
:
A bounded bilinear map f : E × F → G defines a continuous linear map
f : E →L[𝕜] F →L[𝕜] G.
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
IsBoundedBilinearMap.isBigO
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
:
theorem
IsBoundedBilinearMap.isBigO_comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
{α : Type u_5}
(H : IsBoundedBilinearMap 𝕜 f)
{g : α → E}
{h : α → F}
{l : Filter α}
:
theorem
IsBoundedBilinearMap.isBigO'
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
:
theorem
IsBoundedBilinearMap.map_sub_left
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
{x : E}
{y : E}
{z : F}
:
theorem
IsBoundedBilinearMap.map_sub_right
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
{x : E}
{y : F}
{z : F}
:
theorem
IsBoundedBilinearMap.continuous
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
:
Useful to use together with Continuous.comp₂.
theorem
IsBoundedBilinearMap.continuous_left
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
{e₂ : F}
:
Continuous fun e₁ => f (e₁, e₂)
theorem
IsBoundedBilinearMap.continuous_right
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
{e₁ : E}
:
Continuous fun e₂ => f (e₁, e₂)
theorem
ContinuousLinearMap.continuous₂
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(f : E →L[𝕜] F →L[𝕜] G)
:
Continuous (Function.uncurry fun x y => ↑(↑f x) y)
Useful to use together with Continuous.comp₂.
theorem
IsBoundedBilinearMap.isBoundedLinearMap_left
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
(y : F)
:
IsBoundedLinearMap 𝕜 fun x => f (x, y)
theorem
IsBoundedBilinearMap.isBoundedLinearMap_right
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
(x : E)
:
IsBoundedLinearMap 𝕜 fun y => f (x, y)
theorem
isBoundedBilinearMap_smul
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{𝕜' : Type u_5}
[NormedField 𝕜']
[NormedAlgebra 𝕜 𝕜']
{E : Type u_6}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
[NormedSpace 𝕜' E]
[IsScalarTower 𝕜 𝕜' E]
:
IsBoundedBilinearMap 𝕜 fun p => p.fst • p.snd
theorem
isBoundedBilinearMap_mul
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
:
IsBoundedBilinearMap 𝕜 fun p => p.fst * p.snd
theorem
isBoundedBilinearMap_comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
:
IsBoundedBilinearMap 𝕜 fun p => ContinuousLinearMap.comp p.fst p.snd
theorem
ContinuousLinearMap.isBoundedLinearMap_comp_left
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(g : F →L[𝕜] G)
:
IsBoundedLinearMap 𝕜 fun f => ContinuousLinearMap.comp g f
theorem
ContinuousLinearMap.isBoundedLinearMap_comp_right
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
(f : E →L[𝕜] F)
:
IsBoundedLinearMap 𝕜 fun g => ContinuousLinearMap.comp g f
theorem
isBoundedBilinearMap_apply
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
:
IsBoundedBilinearMap 𝕜 fun p => ↑p.fst p.snd
theorem
isBoundedBilinearMap_smulRight
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
:
IsBoundedBilinearMap 𝕜 fun p => ContinuousLinearMap.smulRight p.fst p.snd
The function ContinuousLinearMap.smulRight, associating to a continuous linear map
f : E → 𝕜 and a scalar c : F the tensor product f ⊗ c as a continuous linear map from E to
F, is a bounded bilinear map.
theorem
isBoundedBilinearMap_compMultilinear
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{ι : Type u_5}
{E : ι → Type u_6}
[Fintype ι]
[(i : ι) → NormedAddCommGroup (E i)]
[(i : ι) → NormedSpace 𝕜 (E i)]
:
IsBoundedBilinearMap 𝕜 fun p => ContinuousLinearMap.compContinuousMultilinearMap p.fst p.snd
The composition of a continuous linear map with a continuous multilinear map is a bounded bilinear operation.
def
IsBoundedBilinearMap.linearDeriv
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
(p : E × F)
:
Definition of the derivative of a bilinear map f, given at a point p by
q ↦ f(p.1, q.2) + f(q.1, p.2) as in the standard formula for the derivative of a product.
We define this function here as a linear map E × F →ₗ[𝕜] G, then IsBoundedBilinearMap.deriv
strengthens it to a continuous linear map E × F →L[𝕜] G.
Equations
Instances For
def
IsBoundedBilinearMap.deriv
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
(p : E × F)
:
The derivative of a bounded bilinear map at a point p : E × F, as a continuous linear map
from E × F to G. The statement that this is indeed the derivative of f is
IsBoundedBilinearMap.hasFDerivAt in Analysis.Calculus.FDeriv.
Equations
Instances For
@[simp]
theorem
IsBoundedBilinearMap.deriv_apply
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
(p : E × F)
(q : E × F)
:
↑(IsBoundedBilinearMap.deriv h p) q = f (p.fst, q.snd) + f (q.fst, p.snd)
theorem
ContinuousLinearMap.mulLeftRight_isBoundedBilinear
(𝕜 : Type u_1)
[NontriviallyNormedField 𝕜]
(𝕜' : Type u_5)
[NormedRing 𝕜']
[NormedAlgebra 𝕜 𝕜']
:
IsBoundedBilinearMap 𝕜 fun p => ↑(↑(ContinuousLinearMap.mulLeftRight 𝕜 𝕜') p.fst) p.snd
The function ContinuousLinearMap.mulLeftRight : 𝕜' × 𝕜' → (𝕜' →L[𝕜] 𝕜') is a bounded
bilinear map.
theorem
IsBoundedBilinearMap.isBoundedLinearMap_deriv
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{f : E × F → G}
(h : IsBoundedBilinearMap 𝕜 f)
:
IsBoundedLinearMap 𝕜 fun p => IsBoundedBilinearMap.deriv h p
Given a bounded bilinear map f, the map associating to a point p the derivative of f at
p is itself a bounded linear map.
theorem
Continuous.clm_comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{X : Type u_5}
[TopologicalSpace X]
{g : X → F →L[𝕜] G}
{f : X → E →L[𝕜] F}
(hg : Continuous g)
(hf : Continuous f)
:
Continuous fun x => ContinuousLinearMap.comp (g x) (f x)
theorem
ContinuousOn.clm_comp
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{G : Type u_4}
[NormedAddCommGroup G]
[NormedSpace 𝕜 G]
{X : Type u_5}
[TopologicalSpace X]
{g : X → F →L[𝕜] G}
{f : X → E →L[𝕜] F}
{s : Set X}
(hg : ContinuousOn g s)
(hf : ContinuousOn f s)
:
ContinuousOn (fun x => ContinuousLinearMap.comp (g x) (f x)) s
theorem
Continuous.clm_apply
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{X : Type u_5}
[TopologicalSpace X]
{f : X → E →L[𝕜] F}
{g : X → E}
(hf : Continuous f)
(hg : Continuous g)
:
Continuous fun x => ↑(f x) (g x)
theorem
ContinuousOn.clm_apply
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
{X : Type u_5}
[TopologicalSpace X]
{f : X → E →L[𝕜] F}
{g : X → E}
{s : Set X}
(hf : ContinuousOn f s)
(hg : ContinuousOn g s)
:
ContinuousOn (fun x => ↑(f x) (g x)) s
The set of continuous linear equivalences between two Banach spaces is open #
In this section we establish that the set of continuous linear equivalences between two Banach spaces is an open subset of the space of linear maps between them.
theorem
ContinuousLinearEquiv.isOpen
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
[CompleteSpace E]
:
theorem
ContinuousLinearEquiv.nhds
{𝕜 : Type u_1}
[NontriviallyNormedField 𝕜]
{E : Type u_2}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{F : Type u_3}
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
[CompleteSpace E]
(e : E ≃L[𝕜] F)
: